Sunday, June 13, 2010

Some Reflections on the Mirror Problem

[retrieved from a long-lost backup of a 1998 proto-blog - I think it still makes sense]

Introduction: Reader, meet Problem

If you are human, there is a pretty good chance that you will at some point have looked at people or writing in a mirror and wondered why they are reversed from left to right but not from top to bottom. If you've ever discussed it with friends, you may have come up with ideas ranging from fact that our eyes are side by side to the fact that left and right are relative whereas up and down are absolute.

I recently read an explanation for this phenomenon in an otherwise excellent and best-selling maths popularisation book, and came away with my head hurting and a feeling of dissatisfaction. This wasn't, perhaps, surprising - some years ago a letter about the mirror problem provoked a extraordinary cascade of correspondence in the New Scientist  - everyone appeared to have an answer, but no one had an answer that was so obviously correct that people couldn't help but agree with it.

After reading this book I started thinking about the problem while running. I began to feel that the answer was in some senses rather simple, just terribly counter-intuitive. What a good solution would need was an excellent metaphor, so that you not only ended up knowing the explanation, but knowing that you knew the correct explanation, so clearly that you might even end up wondering why it was ever a problem. I eventually reached such a state - I now clutch the memory of the recent book, of the way-back correspondence, as proof that the problem was ever un-obvious enough to be worth solving.

Here's my solution in three simple steps, with a bit of help from some Friends. And some virtual illustrations. I hope you find it as compelling as I do.

Step 1: Twos and Threes

Width, height, depth. We live in three dimensions. And we have an orientation in each. Picture yourself standing on a giant compass symbol. Your head points up, your face looks North and, should you raise your arms, your right hand would point East and your left West. Turn around, 180 degrees. Your head still points up, but you face South instead of North and your right hand now points West instead of East. Hmmm.. so you've changed your orientation in two out of the three dimensions. OK, back to facing North. Since this is just a thought experiment, let's gently do a cartwheel next, but stop once we're upside down... well, we're still facing North,  but this time our left-right and up-down have been reversed. Back to where we were, and now a simple forward headstand. Now our left-right remains the same, but up-down and front-back have been reversed. So each of the turns involves changing your orientation in two dimensions while you spin round an axis in the third dimension.

The funny thing is, in the real world it doesn't matter how many turns you do, on which axes, in which order, you always end up reversed in exactly two of them, or back where you started and not reversed at all.
Suppose we're chatting at a party and we're standing face to face, and compare our orientation in the three dimensions. Both our bodies are head-up, but my front-back is pointing the opposite way to yours, and my left-right is also reversed compared to yours. It's the same difference as if you had simply turned round in the first of our compass manoeuvres.

In fact, think about anything with a front and a back, a top and a bottom, and a left and a right. A person, a book, even a car. If it's facing you then it will be pointing in the reverse direction to you along not one but precisely two of its dimensions. Front-back (by definition since it's facing you) and either of the two remaining dimensions, up-down or left-right.

Step 2: Ones

Picture yourself in front of a mirror. Compared to you, your image is reversed in just one dimension, front-back. The head still points up, the right hand still points to the East. Right? Mirrors reverseone dimension at a time. It doesn't have to be the front-back dimension, of course, so you can have two mirrors side by side to give an image that looks as if you had just stepped forward and turned around.Is this the answer? It seems simple enough, but why are left with this deep conviction that left and right have been reversed but up and down haven't? Is up-down in some way different from left-right, or is there another explanation?

Step 3: Planet Fussball

Do you know those table-top football (soccer) games that you get in French bars, British student unions and on the US TV series Friends (from where, I must admit, the image popped into my memory in my hour of need)? Where each player stands on one side of the machine spinning the rods with their team's model footballers to score or save goals? Look at the goalie. He is the only figure on his rod, which runs from his right (where you grasp its handle) to his left. We're going to forget about the fact that you can slide him right and left, and just note that the only spin he can do is round his left-right axis, as if he was doing forward or backward somersaults.

Right, now you are a fussball figure. Even weirder, you are in orbit around planet Fussball. (Being moulded from solid plastic, the vacuum fortunately has no ill-affects on you.) You are not alone. The planet is ringed, like Saturn or Jupiter. You are part of a ring, consisting entirely of fellow fussball figures, all of them (like you) born facing in the direction that the ring spins, with their feet towards the planet. Being fussball figures you can, naturally, spin head-over-heels around your waist. Equally naturally you can't spin from left to right, or do cartwheel type spins.

So how do you chat to your friends? You slide up to them and then spin till one of you is upside down (and back to front) relative to the other. In other words the only way you ever see another face is if it is upside-down compared to you, but with the left-right the same as you (unlike this planet where we normally see people's faces  with the top and bottom facing in the same direction as us, but the left-to-right reversed compared to us). Now someone hands you a mirror, and you look at yourself. Being a smart and inquisitive fussball figure you ask yourself "Why is this mirror image reversed top to bottom, but not left to right?". Maybe you wonder if it has anything to do with the fact that up and down are absolutes whereas left and right are relative, or to do with the fact that you have two eyes side by side...

Conclusion: tying it all up

So what does it all prove?

We've established that there's a universal law, that any time you look at something solid face to face, not only is its front-back axis pointing in the opposite direction to yours, but its orientation in one of the other dimensions, either left-right or up-down, will be reversed relative to yours too.

We've noted that on this planet almost anything that turns round, turns round its up-down axis. (Apart from fussball figures, I simply can't think of another example of something with a distinguishable top and bottom, left and right, and back and front that turns, somersault style, around its left-right axis.) So on this planet anything you look at face to face will have its left and right reversed compared to your left and right. In fact this is so common that we don't really think of them as being reversed.

We've noted that we can imagine a planet Fussball where, unlike this planet, it is more common for things and people to turn around their left-right axis than round their up-down axis. We've assumed that people there would take this up-down reversal equally for granted.

We've established that mirror images are only reversed in one dimension, front-back.
Therefore the left-right orientation of your reflection is the same as your left-right orientation, but the opposite of 99% of what you see face to face on this planet, and therefore notable. The up-down orientation is also the same as yours, but being the  same as everything else that you see, not notable.

The simple and complete answer to the mirror problem is that the mirror doesn't mysteriously reverse left and right (or up and down). We reverse left and right on almost everything we see face to face, by spinning it or ourselves round the vertical axis. That's where the mysterious difference between up-down and left-right comes in.

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